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BICSBath Institute for Complex SystemsFinite Element Error Analysis of Elliptic PDEs with RandomCoefficients and its Application to Multilevel Monte Carlo MethodsJulia Charrier, Robert Scheichl and Aretha TeckentrupBath Institute For Complex SystemsPreprint 2/11 (2011)http://www.bath.ac.uk/math-sci/BICS

Finite Element Error Analysis of Elliptic PDEs with RandomCoefficients and its Application to Multilevel Monte Carlo MethodsJulia Charrier ∗ , Robert Scheichl † and Aretha L. Teckentrup †AbstractWe consider a finite element approximation of elliptic partial differential equations with randomcoefficients. Such equations arise, for example, in uncertainty quantification in subsurfaceflow modelling. Models for random coefficients frequently used in these applications, such aslog-normal random fields with exponential covariance, have only very limited spatial regularity,and lead to variational problems that lack uniform coercivity and boundedness with respect tothe random parameter. In our analysis we overcome these challenges by a careful treatmentof the model problem almost surely in the random parameter, which then enables us to proveuniform bounds on the finite element error in standard Bochner spaces. These new boundscan then be used to perform a rigorous analysis of the multilevel Monte Carlo method for theseelliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude,we give some numerical results that confirm the new bounds.Keywords: PDEs with stochastic data, non-uniformly elliptic, non-uniformly bounded, lack offull regularity, log-normal coefficients, truncated Karhunen-Loève expansion.1 IntroductionPartial differential equations (PDEs) with random coefficients are commonly used as models forphysical processes in which the input data are subject to uncertainty. It is often of great importanceto quantify the uncertainty in the outputs of the model, based on the information that is availableon the uncertainty of the input data.In this paper, we consider elliptic PDEs with random coefficients, as they arise for example insubsurface flow modelling (see e.g. [11], [10]). The classical equations governing a steady state,single phase subsurface flow consist of Darcy’s law coupled with an imcompressibility condition.Taking into account the uncertainties in the source terms f and the permeability a of the medium,this leads to a linear, second-order elliptic PDE with random coefficient a(ω, x) and random righthand side f(ω, x), subject to appropriate boundary conditions.Solving equations like this numerically can be challenging for several reasons. Models typicallyused for the coefficient a(ω, x) in applications can vary on a fine scale and have relatively largevariances, meaning that in many cases we only have very limited spatial regularity. In the contextof subsurface flow modelling, for example, a model frequently used for the permeability a(ω, x) isa homogeneous log-normal random field. That is a(ω, x) = exp[g(ω, x)], where g is a Gaussianrandom field. We show in §2.3 that for common choices of mean and covariance functions, inparticular an exponential covariance function for g, trajectories of this type of random field are∗ Département Mathématiques, ENS Cachan, Antenne de Bretagne, Ave. Robert Schumann, 35170 Bruz, France;and INRIA Rennes Bretagne Atlantique (Équipe SAGE). Email: julia.charrier@bretagne.ens-cachan.fr† Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK. Email:r.scheichl@bath.ac.uk and a.l.teckentrup@bath.ac.uk1

only Hölder continuous with exponent less than 1/2. Another difficulty sometimes associated withPDEs with random coefficients, is that the coefficients cannot be bounded uniformly in the randomparameter ω. In the worst case, this leads to elliptic differential operators which are not uniformlycoercive or uniformly bounded. This is for example true for log-normal random fields a(ω, x). Dueto the nature of Gaussian random variables, a(ω, x) can in this case not be bounded from aboveor away from zero uniformly in ω. It is, however, possible to bound a(ω, x) for each fixed ω.In this paper, we consider a finite element approximation (in space) of elliptic PDEs withrandom coefficients as described above, with particular focus on the cases where the coefficienta(ω, x) can not be bounded from above or away from zero uniformly in ω, and trajectories ofa(ω, x) are only Hölder continuous. Indeed, if one assumes that the random coefficient a(ω, x)is sufficiently regular and can be bounded from above and away from zero uniformly in ω, theresulting variational problem is uniformly coercive and bounded, and the well-posedness of theproblem and the subsequent error analysis are classical, see eg. [2, 12, 1, 3]. We here derivebounds on the finite element error in the solution in both L p (H0 1(D)) and Lp (L 2 (D)) norms. Ourerror estimate crucially makes use of the observation that for each fixed ω, we have a uniformlycoercive and bounded problem (in x). The derivation of the error estimate is then based onan elliptic regularity result for coefficients supposed to be only Hölder continuous, making thedependence of all constants on a(ω, x) explicit. We emphasise that we work in standard Bochnerspaces with the usual test and trial spaces as in the deterministic setting. As such, our workbuilds on and complements [13, 5, 17, 9] which all are concerned with the well-posedness andnumerical approximation of elliptic PDEs with infinite dimensional stochastic coefficients that arenot uniformly bounded from above and below (e.g. log-normal coefficients).Finally, applying the new finite element error bounds, we quantify the error committed in themultilevel Monte Carlo (MLMC) method. The MLMC analysis is motivated by [7], where theauthors recently demonstrated numerically the effectiveness of MLMC estimators for computingmoments of various quantities of interest related to the solution u(ω, x) of an elliptic PDE withlog-normal random coefficients. The MLMC method is a very efficient variance reduction techniquefor classical Monte Carlo, especially in the context of differential equations with stochastic dataand stochastic differential equations. It was first introduced by Heinrich [19] for the computationof high-dimensional, parameter-dependent integrals, and has been analysed extensively by Giles[16, 15] in the context of stochastic differential equations in mathematical finance. Under theassumptions of uniform coercivity and boundedness, as well as full regularity, convergence resultsfor the MLMC estimator for elliptic PDEs with random coefficients have recently also been provedin [3]. Here we consider the problem without these assumptions which is of particular interest insubsurface flow applications, where log-normal coefficients are most commonly used.The outline of the rest of this paper is as follows. In §2, we present our model problemand assumptions, and recall from [5], the main results on the error resulting from truncatingthe Karhunen-Loéve expansion in the context of log-normal coefficients. §3 is devoted to theestablishment of the finite element error bound. We first prove a regularity result for elliptic PDEswith Hölder continuous coefficients, making explicit the dependence of the bound on ω. The fullproof is given in the appendix. From this regularity result we then deduce the finite element errorbound in the H 1 -seminorm, before also treating the case of the L 2 -norm and the contribution ofthe quadrature error. We also improve the bound given in [5] for the finite element error in the caseof truncated KL-expansions of log-normal random fields, ensuring uniformity in K (the numberof terms of the truncated expansion). In §4, we use the finite element error analysis from §3 tofurnish a complete error analysis of the multilevel Monte Carlo method. To begin with, we presentbriefly the multilevel Monte-Carlo method for elliptic PDEs with random coefficients proposed in[7], before providing a rigorous bound for the ε-cost. Finally, in §5 we present some numericalexperiments, illustrating the sharpness of some of the results.2

2 Preliminaries2.1 NotationGiven a probability space (Ω, A, P) and a bounded Lipschitz domain D ⊂ R d , we introduce thefollowing notation. For any k ∈ N, we define on the Sobolev space H k (D) the following semi-normand norm:( ∫∑1/2 ( ∫∑1/2|v| H k (D) = |D α v| dx) 2 and ‖v‖ H k (D) = |D α v| dx) 2 .D|α|=kD|α|≤kWe recall that, since D is bounded, the semi-norm | · | H k (D) defines a norm equivalent to the norm‖ · ‖ H k (D) on the subspace H0 k(D) of Hk (D). For any real r ≥ 0, with r /∈ N, set r = k + s withk ∈ N and 0 < s < 1, and denote by | · | H r (D) and ‖ · ‖ H r (D) the Sobolev–Slobodetskii semi-normand norm, respectively, defined for v ∈ H k (D) by( ∫∫|v| H r (D) =D×D∑|α|=k[D α v(x) − D α v(y)] 2|x − y| d+2s dx dy) 1/2 (1/2and ‖v‖ H r (D) = ‖v‖ 2 H k (D) +|v|2 H (D)) r .The Sobolev space H r (D) is then defined as the space of functions v in H k (D) such that theintegral |v| 2 H r (D) is finite. For 0 < s ≤ 1, the space H−s (D) denotes the dual space to H0 s (D) withthe dual norm.In addition to the above Sobolev spaces, we also make use of the Hölder spaces C t (D), with0 < t < 1, on which we define the following semi-norm and norm|v| C t (D) = sup |v(x) − v(y)|x,y∈D: x≠y|x − y| t and ‖v‖ C t (D) = sup |v(x)| + |v| C t (D) .x∈DThe spaces C 0 (D) and C 1 (D) are as usual the spaces of continuous and continuously differentiablefunctions with the standard norms.Finally, we will also require spaces of Bochner integrable functions. To this end, let B be aBanach space with norm ‖ · ‖ B , and v : Ω → B be strongly measurable. With the norm ‖ · ‖ L p (Ω,B)defined by{(∫‖v‖ L p (Ω,B) =Ω ‖v‖p B dP) 1/p , for p < ∞,esssup ω∈Ω ‖v‖ B , for p = ∞,the space L p (Ω, B) is defined as the space of all strongly measurable functions on which this normis finite. In particular, we denote by L p (Ω, H k 0 (D)) the Bochner space where the norm on Hk 0 (D)is chosen to be the seminorm | · | H k (D). For simplicity we write L p (Ω) for L p (Ω, R).The key task in this paper is keeping track of how the constants in the bounds and estimatesdepend on the coefficient a(ω, x) and on the mesh size h. Hence, we will almost always bestating constants explicitly. To simplify this task we shall denote constants appearing in Theorem/Proposition/Corollaryx.x by C x.x . Constants that do not depend on a(ω, x) or h will not beexplicitly stated. Instead, we will write b c for two positive quantities b and c, if b/c is uniformlybounded by a constant independent of a(ω, x) and of h.2.2 Problem SettingWe consider the following linear elliptic partial differential equation (PDE) with random coefficients:−∇ · (a(ω, x)∇u(ω, x)) = f(ω, x), in D, (2.1)u(ω, x) = 0, on ∂D,3

for almost all ω ∈ Ω. The differential operators ∇· and ∇ are with respect to x ∈ D. Let usformally define, for all ω ∈ Ω,a min (ω) := minx∈Da(ω, x) and a max (ω) := max a(ω, x). (2.2)x∈DWe make the following assumptions on the input random field a and on the source term f:A1. a min ≥ 0 almost surely and 1/a min ∈ L p (Ω), for all p ∈ (0, ∞).A2. a ∈ L p (Ω, C t (D)), for some 0 < t ≤ 1 and for all p ∈ (0, ∞).A3. f ∈ L p∗ (Ω, H t−1 (D)), for some p ∗ ∈ (0, ∞].The Hölder continuity of the trajectories of a in Assumption A2 implies that both quantities in(2.2) are well defined, that a max ∈ L p (Ω) and (together with Assumption A1) that a min (ω) > 0and a max (ω) < ∞, for almost all ω ∈ Ω. We will here not make the assumption that we can bounda min (ω) away from zero and a max (ω) away from infinity, uniformly in ω, as this is not true forlog-normal fields a, for example. Many authors work with such assumptions of uniform ellipticityand boundedness. We shall instead work with the quantities a min (ω) and a max (ω) directly.Note also that our assumptions on the (spatial) regularity of the coefficient function a aresignificantly weaker than what is usually assumed in the literature. Most other analyses of thisproblem assume at least that a has a gradient that lies in L ∞ (D). As we will see below, we couldeven weaken Assumptions A1 and A2 and only assume that ‖a‖ C t (D) and 1/a min have a finitenumber of bounded moments, i.e. 0 < p ≤ p a , for some fixed p a > 0, but in order not to complicatethe presentation we did not choose to do this.As usual in finite element methods, we will study the PDE (2.1) in weak (or variational) form,for fixed ω ∈ Ω. This is not possible uniformly in Ω, but almost surely. In the following we will notexplicitly write this each time. With f(ω, ·) ∈ H t−1 (D) and 0 < a min (ω) ≤ a(ω, x) ≤ a max (ω) < ∞,for all x ∈ D, the variational formulation of (2.1), parametrised by ω ∈ Ω, isb ω(u(ω, ·), v)= Lω (v) , for all v ∈ H 1 0 (D), (2.3)where the bilinear form b ω and the linear functional L ω (both parametrised by ω ∈ Ω) are definedas usual, for all u, v ∈ H0 1 (D), by∫b ω (u, v) := a(ω, x)∇u(x) · ∇v(x) dx and L ω (v) := 〈f(ω, ·), v〉 H t−1 (D),H 1−t0 (D) . (2.4)DWe say that for any ω ∈ Ω, u(ω, ·) is a weak solution of (2.1) iff u(ω, ·) ∈ H0 1 (D) and satisfies (2.3).The following result is classical. It is based on the Lax-Milgram Lemma [18].Lemma 2.1. For almost all ω ∈ Ω, the bilinear form b ω (u, v) is bounded and coercive in H0 1(D)with respect to | · | H 1 (D), with constants a max (ω) and a min (ω), respectively. Moreover, there existsa unique solution u(ω, ·) ∈ H0 1 (D) to the variational problem (2.3) and|u(ω, ·)| H 1 (D) ‖f(ω, ·)‖ H t−1 (D).a min (ω)The following proposition is a direct consequence of Lemma 2.1.Theorem 2.2. The weak solution u of (2.1) is unique and belongs to L p (Ω, H 1 0 (D)), for all p < p ∗.Proof. First note that u : Ω → H0 1 (D) is measurable, since u is a continuous function of a.The result then follows directly from Lemma 2.1, Assumptions A1 and A3 and from the Hölderinequality.However, as usual in finite element methods we will require more (spatial) regularity of thesolution u to be able to show convergence. We will come back to this in Section 3.1. First let uslook at some typical examples of input random fields used in applications.4

2.3 Log-normal Random FieldsA coefficient a(ω, x) of particular interest in applications of (2.1) is a log-normal random field,where a(ω, x) = exp [g(ω, x)] with g : Ω × D → R denoting a Gaussian field. We consider onlymean zero homogeneous Gaussian fields with Lipschitz continuous covariance kernel[]C(x, y) = E (g(ω, x)−E[g(ω, x)])(g(ω, y)−E[g(ω, y)]) = k ( ‖x−y‖ ) , for some k ∈ C 0,1 (R + ) (2.5)and for some norm ‖ · ‖ in R d . In particular, this may be the usual modulus ‖x‖ = |x| := (x T x) 1/2or the 1-norm ‖x‖ = ‖x‖ 1 := ∑ di=1 |x i|. A typical example of a covariance function used in practicethat is only Lipschitz continuous is the exponential covariance function given byk(r) = σ 2 exp(−r/λ), (2.6)for some parameters σ 2 (variance) and λ (correlation length).With this type of covariance function, it follows from Kolmogorov’s Theorem [8] that, for allt < 1/2, the trajectories of g belong to C t (D) almost surely. More precisely, Kolmogorov’s Theoremensures the existence of a version ˜g of g (i.e. for any x ∈ D, we have g(·, x) = ˜g(·, x) almost surely)such that ˜g(ω, ·) ∈ C t (D), for almost all ω ∈ Ω. In particular, we have for almost all ω, thatg(ω, ·) = ˜g(ω, ·) almost everywhere. We will identify g with ˜g in what follows.Built on the Hölder continuity of the trajectories of g and using Fernique’s Theorem [8], it wasshown in [5] that Assumption A1 holds and that a ∈ L p (Ω, C 0 (D)), for all p ∈ (0, ∞).Lemma 2.3. Let g be Gaussian with covariance (2.5). Then the trajectories of the log-normal fielda = exp g belong to C t (D) almost surely, for all t < r, and()‖a(ω)‖ C t ≤ 1 + 2 |g(ω)| C t a max (ω) .Proof. Fix ω ∈ Ω and t < 1/2. Since the trajectories of g belong to C t (D) almost surely, we have(|e g(ω,x) − e g(ω,y) | ≤ |g(ω, x) − g(ω, y)| e g(ω,x) + e g(ω,y)) ≤ 2 a max (ω) |g(ω)| C t |x − y| t .for any x, y ∈ D. Now, a max (ω) |g(ω)| C t < ∞ almost surely, and so the result follows by taking thesupremum over all x, y ∈ D.Lemma 2.3 can in fact be generalised from the exponential function to any smooth function of g.Proposition 2.4. Let g be a mean zero Gaussian field with covariance (2.5). Then AssumptionsA1–A2 are satisfied for the log-normal field a = exp g with any t < 1 2 .Proof. Clearly by definition a min ≥ 0. The proof that 1/a min ∈ L p (Ω), for all p ∈ (0, ∞), is basedon an application of Fernique’s Theorem [8] and can be found in [5, Proposition 2.2]. To proveAssumption A2 note that, for all t < 1/2 and p ∈ (0, ∞), g ∈ L p (Ω, C t (D)) (cf. [5, Proposition 3.5])and a max ∈ L p (Ω) (cf. [5, Proposition 2.2]). Thus the result follows from Lemma 2.3 and anapplication of Hölder’s inequality.Smoother covariance functions, such as the Gaussian covariance kernelk(r) = σ 2 exp(−r 2 /λ 2 ), (2.7)which is analytic on D × D, or more generally the covariance functions in the Matérn class withν > 1, all lead to g ∈ C 1 (D) and thus Assumption A2 is satisfied for all t ≤ 1.Remark 2.5. The results in this section can be extended to log-normal random fields for whichthe underlying Gaussian field g(ω, x) does not have mean zero, under the assumption that thismean is sufficiently regular. Adding a mean c(x) to g, we have a(ω, x) = exp[c(x)] exp[g(ω, x)], andassumptions A1-A2 can still be satisfied. In particular, the assumptions still hold if c(x) ∈ C 1/2 (D).5

2.4 Truncated Karhunen-Loève ExpansionsA starting point for many numerical schemes for PDEs with random coefficients is the approximationof the random field a(ω, x) as a function of a finite number of random variables, a(ω, x) ≈a(ξ 1 (ω), ..., ξ K (ω), x). This is true, for example, for the stochastic collocation method discussed inSection 3.4. Sampling methods, such as Monte–Carlo type methods discussed in Section 4, do notrely on such a finite–dimensional approximation as such, but may make use of such approximationsas a way of producing samples of the input random field.A popular choice to achieve good approximations of this kind for log-normal random fields isthe truncated Karhunen-Loève (KL) expansion. For the random field g (as defined in Section 2.2),the KL–expansion is an expansion in terms of a countable set of independent, standard Gaussianrandom variables {ξ n } n∈N . It is given byg(ω, x) =∞∑ √θn b n (x)ξ n (ω),n=1where {θ n } n∈N are the eigenvalues and {b n } n∈N the corresponding normalised eigenfunctions of thecovariance operator with kernel function C(x, y) defined in (2.5). For more details on its derivationand properties, see e.g. [14]. We will here only mention that the eigenvalues {θ n } n∈N are allnon–negative with ∑ n≥0 θ n < +∞.We shall write a(ω, x) as[ ∞]∑ √a(ω, x) = exp θn b n (x)ξ n (ω) ,n=1and denote the random fields resulting from truncated expansions by[K∑ √ K]∑ √g K (ω, x) := θn b n (x)ξ n (ω) and a K (ω, x) := exp θn b n (x)ξ n (ω) , for some K ∈ N.n=1The advantage of writing a(ω, x) in this way is that it gives an expression for a(ω, x) in terms ofindependent, standard Gaussian random variables.Finally, we denote by u K the solution to our model problem with the coefficient replaced by itstruncated approximation,−∇ · (a K (ω, x)∇u K (ω, x)) = f(ω, x), (2.8)subject to homogeneous Dirichlet boundary conditions u K = 0 on ∂D.Under certain additional assumptions on the eigenvalues and eigenfunctions of the covarianceoperator, it is possible to bound the truncation error ‖u − u K ‖ L p (Ω,H 1 0 (D)), for all p < p ∗, as shownin [5]. We make the following assumptions on (θ n , b n ) n≥0 :B1. The eigenfunctions are continuously differentiable, i.e. b n ∈ C 1 (D) for any n ≥ 0.B2. There exists an r ∈ (0, 1) such that,∑θ n ‖b n ‖ 2(1−r)L ∞ (D) ‖∇b n‖ 2rL ∞ (D) < +∞.n≥0Proposition 2.6. Let Assumptions B1–B2 be satisfied, for some r ∈ (0, 1). Then AssumptionsA1–A2 are satisfied also for the truncated KL–expansion a K of a, for any K ∈ N and t < r.Moreover, ‖a K ‖ L p (Ω, C t ( ¯D)) and ‖1/aminK‖ L p (Ω) can be bounded independently of K. If, moreover,Assumption B2 is satisfied for ∂bn∂x iinstead of b n , for all i = 1, . . . , d, then ‖a K ‖ L p (Ω, C 1 ( ¯D)) can bebounded independently of K.6n=1

Proof. This is essentially [5, Propositions 3.7 and 3.8]. The Hölder continuity a K ∈ L p (Ω, C t (D),for all p ∈ (0, ∞), can be deduced as in Lemma 2.3 from the almost sure Hölder continuity of thetrajectories of g K proved in [5, Proposition 3.5].Let us recall the main result on the strong convergence of u to u K from [5, Theorem 4.2].Theorem 2.7. Let Assumptions B1–B2 be satisfied, for some r ∈ (0, 1). Then, u K converges, forall p < p ∗ , to u ∈ L p (Ω, H0 1 (D)). Moreover, for any s ∈ [0, 1], we have‖u − u K ‖ L p (Ω,H 1 0 (D)) ( ∑n>K) 1/2θ n ‖b n ‖ 2(1−r)L ∞ (D) ‖∇b n‖ 2rL ∞ (D)} {{ }=: C 2.7 (K)‖f‖ L p (Ω,H s−1 (D))The hidden constant depends only on D, r, p. Similarly ‖u−u K ‖ L p (Ω,L 2 (D)) C 2.7 ‖f‖ L p (Ω,H s−1 (D)).Assumptions B1–B2 are fulfilled, among other cases, for the analytic covariance function (2.7)as well as for the exponential covariance function (2.6) with 1-norm ‖x‖ = ∑ di=1 |x i| in (2.5),since then the eigenvalues and eigenvectors can be computed explicitly and we have explicit decayrates for the KL–eigenvalues. For details see [5, Section 7]. In the latter case, on non-rectangulardomains D, we need to use a KL-expansion on a bounding box containing D to get again explicitformulae for the eigenvalues and eigenvectors. Strictly speaking this is not a KL–expansion on D.Proposition 2.8. We have the following bound on the constant in Theorem 2.7:⎧⎨ K ρ−12 , for the 1-norm exponential covariance kernel,C 2.7 (K) ⎩K d−12d exp ( − c 1 K 1/d) , for the Gaussian covariance kernel,for some constant c 1 > 0 and for any 0 < ρ < 1. The hidden constants depend only on D, ρ, p.3 Finite Element Error AnalysisLet us now come to the central part of this paper. To carry out a finite element error analysisfor (2.1) under Assumptions A1-A3, we will require first of all regularity results for trajectoriesof the weak solution u. However, since we did not assume uniform ellipticity or boundedness ofb ω (·, ·), it is essential that we track exactly, how the constants in the regularity estimates dependon the input random field a. We were unable to find such a result in the literature, and we willtherefore in Section 3.1 reiterate the steps in the usual regularity proof for the solution u(ω, ·) of(2.3) following the proof in Hackbusch [18], but highlighting explicitly where and how constantsdepend on a. A detailed proof is given in Appendix A. We will need to assume that D ⊂ R d is aC 2 bounded domain for this proof. Regularity proofs that require only Lipschitz continuity for theboundary of D do exist (cf. Hackbusch [18, Theorems 9.1.21 and 9.1.22]), but we did not considerit instructive to complicate the presentation even further. We do not see any obstacles in alsogetting an explicit dependence of the constants in the case of Lipschitz continuous boundaries.Having established the regularity of trajectories of u, we then carry out a classical finite elementerror analysis for each trajectory in Section 3.2 and deduce error estimates for the moments of theerror in u. Since we only have a regularity result for C 2 bounded domains and since the integralsappearing in b ω (·, ·) can in general only be approximated by quadrature, we will also need to addressvariational crimes, such as the approximation of a C 2 bounded domain by a polygonal domain aswell as quadrature errors (cf. Section 3.3).Let us assume for the rest of this section that D ⊂ R d is a C 2 bounded domain.7

3.1 Regularity of the SolutionProposition 3.1. Let Assumptions A1-A3 hold with 0 < t < 1. Then, u(ω, ·) ∈ H 1+s (D), for all0 < s < t except s = 1/2 almost surely in ω ∈ Ω, and‖u(ω, ·)‖ H 1+s (D) C 3.1 (ω) ‖f(ω, ·)‖ H s−1 (D), where C 3.1 (ω) := a max(ω)‖a(ω, ·)‖ C t (D)a min (ω) 3 .If the assumptions hold with t = 1, then u(ω, ·) ∈ H 2 (D) and ‖u(ω, ·)‖ H 2 (D) C 3.1 (ω) ‖f(ω, ·)‖ L 2 (D).We give here only the main elements of the proof and consider only the case t < 1 in detail. Itfollows the proof of [18, Theorem 9.1.16] and consists in three main steps. We formulate the firsttwo steps as separate lemmas and then give the final step following the lemmas. We fix ω ∈ Ω andto simplify the notation we will not specify the dependence on ω anywhere in the proof.In the first step of the proof we take D = R d and establish the regularity of a slightly moregeneral elliptic PDE with tensor-valued coefficients.Lemma 3.2. Let 0 < t < 1 and D = R d , and let A = (A ij ) d i,j=1 ∈ S d(R) be a symmetric, uniformlypositive definite n × n matrix-valued function from D to R n×n , i.e. there exists A min > 0 such thatA(x)ξ · ξ ≥ A min |ξ| 2 uniformly in x ∈ D and ξ ∈ R d , and let A ij ∈ C t (D), for all i, j = 1, . . . , d.Consider−div(A(x)∇w(x)) = F (x) in D (3.1)with F ∈ H s−1 (R d ), for some 0 < s < t. Any weak solution w ∈ H 1 (R d ) of (3.1) is in H 1+s (R d )and‖w‖ H 1+s (R d ) 1 ()|A|A C t (R d ,S d (R)) |w| H 1 (R d ) + ‖F ‖ H s−1 (R d ) + ‖w‖ H 1 (R d ),minwhere |A| C t (R d ,S d (R)) is the Hölder seminorm on S d (R) using a suitable matrix norm.Proof. This is essentially [18, Theorem 9.1.8] with the dependence on A made explicit, and it canbe proved using the representation of the norm on H 1+s (R d ) via Fourier coefficients, as well as afractional difference operator Rh i , i = 1, . . . , d, on a Cartesian mesh with mesh size h > 0 (similarto the classical Nirenberg translation method for proving H 2 regularity).It is shown in the proof of [18, Theorem 9.1.8] that ∑ di=1 ‖(Ri h )∗ w‖ H 1 (R d ) is an upper bound for‖w‖ H 1+s (R d ) and that it can itself be bounded from above in terms of ‖w‖ H 1 (R d ) and ‖F ‖ H s−1 (R d )using the weak form (2.3) of our problem. The dependence of this upper bound on A min stemsfrom the fact that in order to use (2.3), we have to switch from |(Rh i )∗ w| H 1 (R d ) to the energy norm∫RA(x)|∇(R i d h )∗ w| 2 dx. The dependence on |A ij | C t (R d ) comes from bounding the differences of A ijat two consecutive grid points in the translation step.For a definition of Rh i and more details see [18, Theorem 9.1.8] or Section A.1 in the appendix.The second step consists in treating the case where D = R d + := {y = (y 1 , ..., y d ) : y d > 0}.Lemma 3.3. Let 0 < t < 1 and D = R d +, and let A : D → S d (R) be as in Lemma 3.2. Considernow (3.1) on D = R d + subject to w = 0 on ∂D with F ∈ H s−1 (R d +), for some 0 < s < t, s ≠ 1/2.Then any weak solution w ∈ H 1 (R d +) of this problem is in H 1+s (R d +) and‖w‖ H 1+s (R d + ) A )max(|A|A 2 C t (R d +min,S d(R)) |w| H 1 (R d + ) + ‖F ‖ H s−1 (R d + ) + A max‖w‖A H 1 (R d min+ ) .where A max := ‖A‖ C 0 (R d + ,S d(R)) . 8

Proof. This is essentially [18, Theorem 9.1.11] with the dependence on A made explicit. It usesthe fact that for any s > 0, s ≠ 1/2, the normd∑∥|||v||| s :=(‖v‖ 2 L 2 (R d + ) + ∥ ∂v ∥∥2∂x ii=1H s−1 (R d + ) ) 1/2is equivalent to the usual norm on H s (R d +). Then using the same approach as in the proof ofLemma 3.2, we can establish that, for 1 ≤ j ≤ d − 1,∥ ∂w ∥∥∥H∥ 1 ()|A|∂x j s (R d + ) A C t (R d min + ,S d(R)) |w| H 1 (R d + ) + ‖F ‖ H s−1 (R d + ) + ‖w‖ L 2 (R d + ) . (3.2)To establish a similar bound for j = d is technical. We use (3.1) and the following inequality, forany D ⊂ R d and 0 < s < t < 1:‖bv‖ H s (D) |b| C t (D) ‖v‖ L 2 (D) + ‖b‖ C 0 (D) ‖v‖ H s (D), for all b ∈ C t (D), v ∈ H s (D). (3.3)We use (3.3) to bound the H s ∂w–norm of A ij ∂x j, for (i, j) ≠ (d, d), and so by rearranging the weakform of (3.1) we can also bound the H s ∂w–norm of A dd ∂x d. This leads to the additional factor A maxin the bound. The final result can then be deduced by applying (3.3) once more, with b = 1/A dd∂wand v = A dd ∂x d, leading to an additional factor 1/A min . For details see [18, Theorem 9.1.11] orSection A.2 in the appendix.Proof of Proposition 3.1. We are now ready to prove Proposition 3.1 using Lemmas 3.2 and 3.3.The third and last step consists in using a covering of D by m + 1 bounded regions (D i ) 0≤i≤m ,such thatm⋃m⋃D 0 ⊂ D, D ⊂ D i and ∂D = (D i ∩ ∂D).i=0Using a (non-negative) partition of unity {χ i } 0≤i≤m ⊂ C ∞ (R d ) subordinate to this cover, it ispossible to reduce the proof to bounding ‖χ i u‖ H 1+s (D), for all 0 ≤ i ≤ m.For i = 0 this reduces to an application of Lemma 3.2 with w and F chosen to be extensions by0 from D to R d of χ 0 u and of fχ 0 +a∇u·∇χ 0 +div(au∇χ 0 ), respectively. The tensor A degeneratesto a(x)I d , where a is a smooth extension of a(x) on D 0 to a min on R d \D, and so A min = a min and|A| C t (R d ,S d (R)) ≤ |a| C t (D).For 1 ≤ i ≤ m, the proof reduces to an application of Lemma 3.3. As for i = 0, we can seethat χ i u ∈ H0 1(D ∩ D i) is the weak solution of the problem −div(a∇u i ) = f i on D ∩ D i withf i := fχ i + a∇u · ∇χ i + div(au∇χ i ). To be able to apply Lemma 3.3 to the weak form of this PDE,we define now a twice continuously differentiable bijection α i (with αi−1 also in C 2 ) from D i to thecylinderQ i := {y = (y 1 , . . . , y d ) : |(y 1 , . . . , y d−1 )| < 1 and |y d | < 1},such that D i ∩ D is mapped to Q i ∩ R d +, and D i ∩ ∂D is mapped to Q i ∩ {y : y d = 0}. We useαi−1 to map all the functions defined above on D i ∩ D to Q i ∩ R d +, and then extend them suitablyto functions on R d + to finally apply Lemma 3.3. The tensor A in this case is a genuine tensordepending on the mapping α i . However, since ∂D was assumed to be C 2 , we get A min a min ,A max a max and |A| C t (R d + ,S d(R)) |a| C t (R d ), with hidden constants that only depend on α i, αi−1+and their Jacobians. For details see [18, Theorem 9.1.16] or Section A.3 in the appendix.Using Proposition 3.1 and Assumptions A1-A3, we can now conclude on the regularity of u.i=19

Theorem 3.4. Let Assumptions A1-A3 hold with 0 < t ≤ 1. Then u ∈ L p (Ω, H 1+s (D)), for allp < p ∗ and for all 0 < s < t except s = 1/2. If t = 1, then u ∈ L p (Ω, H 2 (D)).Proof. Since a max (ω) ≤ ‖a(ω, ·)‖ C t (D) and Hs−1 (D) ⊂ H t−1 (D), for all s < t, the result followsdirectly from Proposition 3.1 and Assumptions A1–A3 via Hölder’s inequality.Remark 3.5. Note that in order to establish u ∈ L p (Ω, H 1+s (D)), for some fixed p > 0, it wouldhave been sufficient to assume that the constant C 3.1 in Proposition 3.1 is in L q (Ω) for q = p∗ pp . ∗−pIn the case p ∗ = ∞, q = p is sufficient. This in turn implies that we can weaken Assumption A1to 1/a min ∈ L q (Ω) with q > 3p, or Assumption A2 to a ∈ L q (Ω, C t (D)) with q > 2p, or bothassumptions to L q with q > 5p.However, in the case of a log-normal field a and p ∗ = ∞, we do have bounds on all momentsp ∈ (0, ∞), but in general we only have the limited spatial regularity of 1 + s < 3/2.3.2 Finite Element ApproximationWe consider finite element approximations of our model problem (2.1) using standard, continuous,piecewise linear finite elements. The aim is to derive estimates of the finite element error in theL p (Ω, H 1 0 (D)) and Lp (Ω, L 2 (D)) norms. To remain completely rigorous, we keep the assumptionthat boundary of D is C 2 , so that we can apply the explicit regularity results from the previoussection. However, this means that we will have to approximate our domain D by polygonal domainsD h in dimensions d ≥ 2.We denote by {T h } h>0 a shape-regular family of simplicial triangulations of the domain D,parametrised by its mesh width h := max τ∈Th diam(τ), such that, for any h > 0,• D ⊂ ⋃ τ∈T hτ, i.e. the triangulation covers all of D, and• the vertices x τ 1 , . . . , xτ d+1 of any τ ∈ T h lie either all in D or all in R d \D.Let D h denote the union of all simplices that are interior to D and D h its interior, so that D h ⊂ D.Associated with each triangulation T h we define the spaceV h :={v h ∈ C(D) : v h | τ linear, for all τ ∈ T h with τ ⊂ D h ,}and v h | D\Dh= 0 (3.4)of continuous, piecewise linear functions on D h that vanish on the boundary of D h and in D\D h .Let us recall the following standard interpolation result (see e.g. [4, Section 4.4]).Lemma 3.6. Let v ∈ H 1+s (D h ), for some 0 < s ≤ 1. ThenThe hidden constant is independent of h and v.infv h ∈V h|v − v h | H 1 (D h ) ‖v‖ H 1+s (D h ) h s . (3.5)This can easily be extended to an interpolation result for functions v ∈ H 1+s (D) ∩ H0 1 (D), byestimating the residual over D\D h . However, when D is not convex it requires local mesh refinementin the vicinity of any non-convex parts of the boundary. We make the following assumption on T h :A4 For all τ ∈ T h with τ ∩ D h = ∅ and x τ 1 , . . . , xτ d+1 ∈ D, we assume diam(τ) h2 .Lemma 3.7. Let v ∈ H 1+s (D) ∩ H0 1 (D), for some 0 < s ≤ 1, and let Assumption A4 hold. Theninf |v − v h | Hv h ∈V 1 (D) ‖v‖ H 1+s (D) h s . (3.6)h10

Proof. This result is classical (for parts of the proof see [18, Section 8.6] or [20]). Set D δ := D\D hwhere δ denotes the maximum width of D δ , and let first s = 1. Since v h = 0 on D δ it suffices toshow that|v| H 1 (D δ ) ‖v‖ H 2 (D) h . (3.7)The result then follows for s = 1 with Lemma 3.6. The result for s < 1 follows by interpolation,since trivially, |v| H 1 (D δ ) ≤ ‖v‖ H 1 (D),To show (3.7), let w ∈ H 1 (D). Using a trace result we get‖w‖ L 2 (D δ ) ≤ ‖w‖ L 2 (S δ ) δ 1/2 ‖w‖ H 1 (D),where S δ = {x ∈ D : dist(x, ∂D) ≤ δ} ⊂ D is the boundary layer of width δ. It follows fromAssumption A4 that diam(τ) h 2 wherever the boundary is not convex. In regions where D isconvex it follows from the smoothness assumption on ∂D that the width of D δ is O(h 2 ). Henceδ h 2 , which completes the proof of (3.7).Now, for almost all ω ∈ Ω, the finite element approximation to (2.3), denoted by u h (ω, ·), isthe unique function in V h that satisfiesb ω (u h (ω, ·), v h ) = L ω (v h ), for all v h ∈ V h . (3.8)Since b ω (·, ·) is coercive and bounded in H0 1 (D) (cf. Lemma 2.1), we have|u h (ω, ·)| H 1 (D) ‖f(ω, ·)‖ H t−1 (D)/a min (ω). (3.9)as well as the following classical quasi optimality result (cf. [4, 18]).Lemma 3.8 (Cea’s Lemma). Let Assumptions A1–A3 hold. Then, for almost all ω ∈ Ω,|u(ω, ·) − u h (ω, ·)| H 1 (D) ≤ C 3.8 (ω)inf |u(ω, ·) − v h | Hv h ∈V 1 (D), where C 3.8 (ω) :=h( )amax (ω) 1/2.a min (ω)Combining this with the interpolation result above we get the following error estimates.Theorem 3.9. Let Assumptions A1–A4 hold, for some 0 < t < 1 and p ∗ ∈ (0, ∞]. Then, for allp < p ∗ , s < t with s ≠ 1/2 and h > 0, we have‖u − u h ‖ L p (Ω,H0 1(D)) C 3.9 ‖f‖ L p∗(Ω,H s−1 (D)) h s a 3/2max‖a‖ C, where C 3.9 :=t (D)∥ a 7/2 ∥minwith q = p∗ pp ∗−p. If Assumptions A2 and A3 hold with t = 1, then‖u − u h ‖ L p (Ω,H 1 0 (D)) C 3.9 ‖f‖ L p∗(Ω,L 2 (D)) h .∥L q (Ω)Proof. Let 0 < t < 1 in Assumptions A2 and A3. It follows from Proposition 3.1 and Lemmas 3.7and 3.8 that, for almost all ω ∈ Ω,|u(ω, ·) − u h (ω, ·)| H 1 (D) C 3.1 (ω) C 3.8 (ω) ‖f(ω, ·)‖ H s−1 (D) h s . (3.10)The result now follows by Hölder’s inequality. The proof for t = 1 is analogous.The usual duality (or Aubin–Nitsche) “trick” leads to a bound on the L 2 –error.11

Corollary 3.10. Let Assumptions A1–A4 hold, for some 0 < t < 1 and p ∗ ∈ (0, ∞]. Then, for allp < p ∗ , s < t with s ≠ 1/2 and h > 0, we have‖u − u h ‖ L p (Ω,L 2 (D)) C 3.10 ‖f‖ L p∗(Ω,H s−1 (D)) h 2s amax‖a‖ 7/2 2 C, where C 3.10 :=t (D)∥∥with q = p∗ pp ∗−p. If Assumptions A2 and A3 hold with t = 1, then‖u − u h ‖ L p (Ω,L 2 (D)) C 3.10 ‖f‖ L p∗(Ω,L 2 (D)) h 2 .a 13/2min∥L q (Ω)Proof. We will use a duality argument. For almost all ω ∈ Ω, let e ω := u(ω, ·)−u h (ω, ·) and denoteby w ω the solution to the adjoint problemb ω (v, w ω ) = (e ω , v)for all v ∈ H 1 0 (D),which, by Proposition 3.1 with f(ω, ·) = e ω , is also in H 1+s (D). By Galerkin orthogonality‖e ω ‖ 2 L 2 (D) = (e ω, e ω ) L 2 (D) = b ω (e ω , w ω − z h ), for any z h ∈ V h .Using the boundedness of b ω (·, ·) and Lemma 3.7, we then get‖e ω ‖ 2 L 2 (D) a max(ω) |u(ω, ·) − u h (ω, ·)| H 1 (D) ‖w ω ‖ H 1+s (D) h s .Now it follows from Proposition 3.1 and Theorem 3.9 that‖e ω ‖ 2 L 2 (D) ≤ a max(ω) C 3.1 (ω) 2 C 3.8 (ω) ‖f(ω, ·)‖ H s−1 (D)‖e ω ‖ L 2 (D) h 2s .Dividing by ‖e ω ‖ L 2 (D), the result follows again by an application of Hölder’s inequality.Remark 3.11. As usual, the analysis can be extended in a straightforward way to other boundaryconditions, to tensor coefficients, or to more complicated PDEs including low–order terms, providedthe boundary data and the coefficients are sufficiently regular, see [18] for details.3.3 Quadrature ErrorThe integrals appearing in the bilinear form∑∫b ω (w h , v h ) =a(ω, x)∇w h · ∇v h dxττ∈T h : τ∈D hand in the linear functional L ω (v h ) involve realisations of random fields. It will in general beimpossible to evaluate these integrals exactly, and so we will use quadrature instead. We will onlyexplicitly analyse the quadrature error in b ω , but the quadrature error in approximating L ω (v h )can be analysed analogously.In our analysis, we use the midpoint rule, approximating the integrand by its value at themidpoint x τ of each simplex τ ∈ T h . The trapezoidal rule which we use in the numerical sectioncan be analysed analogously. Let us denote the resulting bilinear form that approximates b ω onthe grid T h by∑∫˜bω (w h , v h ) = a(ω, x τ ) ∇w h (x) · ∇v h (x) dx ,ττ∈T h : τ∈D hand let ũ h (ω, ·) denote the corresponding solution to˜bω (ũ h (ω, ·), v h ) = L ω (v h ) , for all v h ∈ V h .Clearly the bilinear form ˜b ω is bounded and coercive, with the same constants as the exactbilinear form b ω and so we can apply the following classical result [6] (with explicit dependence ofthe bound on the coefficients).12

Lemma 3.12 (First Strang Lemma). Let Assumptions A1-A3 hold. Then, for almost all ω ∈ Ω,|u(ω, ·) − ũ h (ω, ·)| H 1 (D) ≤{ (inf 1 + a )max(ω)|u(ω, ·) − v h |v h ∈V h a min (ω)H 1 (D)+1a min (ω)|b ω (v h , w h ) −sup˜b}ω (v h , w h )|.w h ∈V h|w h | H 1 (D)Proposition 3.13. Let Assumptions A1–A4 hold, for some 0 < t < 1 and p ∗ ∈ (0, ∞]. Then, forall p < p ∗ , s < t with s ≠ 1/2 and 0 < h < 1, we have‖u − ũ h ‖ L p (Ω,H0 1(D)) C 3.13 ‖f‖ L p∗(Ω,H s−1 (D)) h s a 5/2max‖a‖ C, where C 3.13 :=t (D)∥ a 9/2 ∥minwith q = p∗ pp ∗−p. If Assumptions A2 and A3 hold with t = 1, thenProof. We first note that, for all w h ∈ V h ,∣∣b ω (v h , w h ) − ˜b ∑ω (v h , w h ) ∣ =∣‖u − ũ h ‖ L p (Ω,H 1 0 (D)) C 3.13 ‖f‖ L p∗(Ω,L 2 (D)) h .≤ ∑τ∈T h∫ττ∈T h∫τ(a(ω, x) − a(ω, x τ )) ∇v h · ∇w h dx∣∥L q (Ω)|a(ω, x) − a(ω, x τ )||x − x τ | t |x − x τ | t |∇v h (ω) · ∇w h | dx≤ |a(ω)| C t (D) ht |v h | H 1 (D) |w h | H 1 (D) .Hence, it follows from Lemma 3.12 that, for almost all ω ∈ Ω,{ (|u(ω, ·) − ũ h (ω, ·)| H 1 (D) ≤ inf 1 + a )max(ω)|u(ω, ·) − v h |v h ∈V h a min (ω)H 1 (D) + h |a(ω)| }t C t (D)|v h |a min (ω) H 1 (D) .Let us now make the particular choice v h := u h (ω, ·) ∈ V h , i.e. the solution of (3.8). Then itfollows from (3.9) and (3.10) and the fact that h t < h s , for any s < t ≤ 1 and h < 1, that( (|u(ω, ·) − ũ h (ω, ·)| H 1 (D) 1 + a )max(ω)C 3.1 (ω) C 3.8 (ω) + |a(ω)| )C t (D)a min (ω)a min (ω) 2 ‖f(ω, ·)‖ H s−1 (D) h s .The result follows again via an application of Hölder’s inequality.Remark 3.14. To recover the O(h 2s ) convergence for the L 2 -error ‖u − ũ h ‖ L p (Ω,L 2 (D)) in thecase of quadrature, we require additional regularity of the coefficient function a. If a(ω, ·) is atleast C 2s , then we can again obtain O(h 2s ) convergence even with quadrature, using duality asin Corollary 3.10. This is for example the case in the context of lognormal random fields withGaussian covariance kernel, where a(ω, ·) ∈ C ∞ (D). In the context of the exponential covariancekernel the L 2 -convergence rate is always bounded by O(h 1/2−δ ), due to the lack of regularity in a.13

3.4 Truncated FieldsCombining the results from Sections 2.4 and 3.2 we can also estimate the error in the case oftruncated KL-expansions of log-normal coefficients which we will use in the numerical experimentsin Section 5. These results give an improvement of the error estimate given in [5] for log-normalcoefficients that do not admit full regularity, e.g. those with exponential covariance kernel. Thenovelty with respect to [5] is the improvement of the bound for the finite element error, leading toa uniform bound with respect to the number of terms in the KL–expansion. We will assume herethat f(ω, ·) ∈ L 2 (D).Let a be a log-normal field as defined in Section 2.4 via a KL expansion and let a K be theexpansion truncated after K terms. Now, for almost all ω ∈ Ω, let u K,h (ω, ·) denote the uniquefunction in V h that satisfiesb K,ω (u K,h (ω, ·), v h ) = L ω (v h ), for all v h ∈ V h , (3.11)where b K,ω (u, v) := ∫ D a K(ω, x)∇u · ∇v dx. Then we have the following result.Theorem 3.15. Let f ∈ L p∗ (Ω, L 2 (D)), for some p ∗ ∈ (0, ∞]. Then, in the Gaussian covariancecase, there exists a c 1 > 0 such that, for any p < p ∗ ,(‖u − u K,h ‖ L p (Ω,H0 1(D)) h + K d−1 (2d exp − c1 K 1/d)) ‖f‖ L p ∗ (Ω,L 2 (D)) .In the case of the 1-norm exponential covariance, for any p < p ∗ , 0 < ρ < 1 and 0 < s < 1/2,()‖u − u K,h ‖ L p (Ω,H0 1(D)) C 3.15 h s + K ρ−12 ‖f‖ L p ∗ (Ω,L 2 (D)) ,where C 3.15 := min{η(K) h 1−s , C 3.9 } and η(K) is an exponential function of K given in [5, Proposition6.3].Proof. It follows from Proposition 2.6 that a K satisfies assumptions A1–A2. Therefore we canapply Theorem 3.9 to bound ‖u K − u K,h ‖ L 2 (Ω,H0 1 (D)). The result then follows from Theorem 2.7,Proposition 2.8 and the triangle inequality.In contrast to [5, Proposition 6.3] this bound is uniform in K. Note that for small values of K,the constant C 3.15 will actually be of order O(h 1−s ) leading to a linear dependence on h also in theexponential covariance case (as stated in [5, Proposition 6.3]). For larger values of K this will notbe the case and the lower regularity of u K will affect the convergence rate with respect to h.Remark 3.16. As we will see in Section 5, in the exponential covariance case for correlationlengths that are smaller than the diameter of the domain, a relatively large number K of KLmodesis necessary to get even 10% accuracy. The new uniform error bound in Theorem 3.15 istherefore crucial also for the analysis of stochastic Galerkin and stochastic collocation methods,such as the one given in [5].4 Convergence Analysis of Multilevel Monte Carlo MethodsWe will now apply this new finite element error analysis in Section 4, to give a rigorous bound onthe cost of the multilevel Monte Carlo method applied to (2.1) for general random fields satisfyingAssumptions A1–A3, and to establish its superiority over the classical Monte Carlo method. Thisbuilds on the recent paper [7]. We start by briefly recalling the classical Monte Carlo (MC) and14

multilevel Monte Carlo (MLMC) algorithms for PDEs with random coefficients, together with themain results on their performance. For a more detailed description of the methods, we refer thereader to [7] and the references therein.In the Monte Carlo framework, we are usually interested in finding the expected value of somefunctional Q = G(u) of the solution u to our model problem (2.1). Since u is not easily accessible, Qis often approximated by the quantity Q h := G(u h ), where u h denotes the finite element solution ona sufficiently fine spatial grid T h . Thus, to estimate E [Q] := ‖Q‖ L 1 (Ω), we compute approximations(or estimators) ̂Q h to E [Q h ], and quantify the accuracy of our approximations via the root meansquare error (RMSE)e( ̂Q(h ) := E [ ( ̂Q h − E(Q)) 2]) 1/2.The computational cost C ε ( ̂Q h ) of our estimator is then quantified by the number of floating pointoperations that are needed to achieve a RMSE of e( ̂Q h ) ≤ ε. This will be referred to as the ε–cost.The classical Monte Carlo (MC) estimator for E [Q h ] iŝQ MCh,N := 1 NN∑Q h (ω (i) ), (4.1)i=1where Q h (ω (i) ) is the ith sample of Q h and N independent samples are computed in total.There are two sources of error in the estimator (4.1), the approximation of Q by Q h , which isrelated to the spatial discretisation, and the sampling error due to replacing the expected valueby a finite sample average. This becomes clear when expanding the mean square error (MSE)and using the fact that for Monte Carlo E[ ̂QMCh,N ] = E[Q h] and V[ ̂QMCh,N ] = N −1 V[Q h ], whereV[X] := E[(X − E[X]) 2 ] denotes the variance of the random variable X : Ω → R. We getMCe( ̂Q h,N )2 = N −1 V[Q h ] + ( E[Q h − Q] ) 2 . (4.2)A sufficient condition to achieve a RMSE of ε with this estimator is that both of these terms areless than ε 2 /2. For the first term, this is achieved by choosing a large enough number of samples,N = O(ε −2 ). For the second term, we need to choose a fine enough finite element mesh T h , suchthat E[Q h − Q] = O(ε).The main idea of the MLMC estimator is very simple. We sample not just from one approximationQ h of Q, but from several. Linearity of the expectation operator implies thatE[Q h ] = E[Q h0 ] +L∑E[Q hl − Q hl−1 ] (4.3)l=1where {h l } l=0,...,L are the mesh widths of a sequence of increasingly fine triangulations T hl withT h := T hL , the finest mesh, and h l−1 /h l ≤ M ∗ , for all l = 1, . . . , L. Hence, the expectation on thefinest mesh is equal to the expectation on the coarsest mesh, plus a sum of corrections adding thedifference in expectation between simulations on consecutive meshes. The multilevel idea is now toindependently estimate each of these terms such that the overall variance is minimised for a fixedcomputational cost.Setting for convenience Y 0 := Q h0 and Y l := Q hl − Q hl−1 , for 1 ≤ l ≤ L, we define the MLMCestimator simply aŝQ L MLh,{N l } := ∑L∑Ŷl,N MC 1 ∑N ll=Y l (ω (i) ), (4.4)l=0N ll=0 i=1where importantly Y l (ω (i) ) = Q hl (ω (i) ) − Q hl−1 (ω (i) ), i.e. using the same sample on both meshes.15

Since all the expectations E[Y l ] are estimated independently in (4.3), the variance of the MLMCestimator is ∑ LV[Y l ] and expanding as in (4.2) leads again tol=0 N −1lMLe( ̂Qh,{N l } )2[ ( ]:= E ̂QML h,{N l } − E[Q]) 2=L∑l=0N −1lV[Y l ] + ( E[Q h − Q] ) 2 . (4.5)As in the classical MC case before, we see that the MSE consists of two terms, the variance of theestimator and the error in mean between Q and Q h . Note that the second term is identical to thesecond term for the classical MC method in (4.2). A sufficient condition to achieve a RMSE of ε isagain to make both terms less than ε 2 /2. This is easier to achieve with the MLMC estimator, as• for sufficiently large h 0 , samples of Q h0 are much cheaper to obtain than samples of Q h ;• the variance Y l tends to 0 as h l → 0, meaning we need fewer samples on T hl , for l > 0.Let now C l denote the cost to obtain one sample of Q hl . Then we have the following results on theε–cost of the MLMC estimator (cf. [7, 16]).Theorem 4.1. Suppose that there are positive constants α, β, γ > 0 such that α≥ 1 2min(β, γ) andM1. |E[Q h − Q]| = O(h α )M2. V[Q hl − Q hl−1 ] = O(h β l )M2. C l = O(h −γl),Then, for any ε γ,⎨MLC ε ( ̂Qh,{N l } ) ε −2 (log ε) 2 , if β = γ,⎪ ⎩ε −2−(γ−β)/α , if β < γ.For the classical MC estimator we have C ε ( ̂QMCh ) = O(ε−2−γ/α ).We can now use the results from Section 3.2 to verify Assumptions M1–M2 in Theorem 4.1for some simple functionals G(u). More complicated functionals could then be tackled by dualityarguments in a similar way.Proposition 4.2. Let Q = G(u) := |u| q H 1 (D) , for some 1 ≤ q < p ∗/2. Suppose that D is a C 2bounded domain and that Assumptions A1–A4 hold, for some 0 < t < 1. Then AssumptionsM1–M2 in Theorem 4.1 hold for any α < t and β < 2t. For t = 1, we get α = 1 and β = 2.Proof. We only give the proof for t < 1. The proof for t = 1 is analogous. Let Q h := |u h | q H 1 (D) .Using the expansion a q − b q = (a − b) ∑ q−1j=0 aj b q−1−j , for a, b ∈ R and q ∈ N, we get{}|Q(ω) − Q h (ω)| |u(ω, ·) − u h (ω, ·)| H 1 (D) max |u(ω, ·)| q−1H 1 (D) , |u h(ω, ·)| q−1H 1 (D) , 1 , almost surely.This also holds for non-integer values of q > 1. Now, it follows from Lemma 2.1 and Theorem 3.9that, for any α < t,|Q(ω) − Q h (ω)| C 3.1(ω) C 3.8 (ω)a q−1min (ω) ‖f(ω, ·)‖ q H α−1 (D) hα , almost surely.16

|u| H 1 (D)‖u‖ L 2 (D)|u| H 13 ε −5 ε −3 ε −7/2 ε −2 3 ε −8 ε −6 ε −5 ε −3‖u‖ L 2 (D)d MC MLMC MC MLMC d MC(D)MLMC MC MLMC1 ε −3 ε −2 ε −5/2 ε −2 1 ε −4 ε −2 ε −3 ε −22 ε −4 ε −2 ε −3 ε −2 2 ε −6 ε −4 ε −4 ε −2Table 1: Theoretical upper bounds for the ε-costs of classical and multilevel Monte Carlo from Theorem4.1 in the case of log-normal fields a with Gaussian covariance function (left) and exponentialcovariance function (right). (For simplicity we wrote ε −p , instead of ε −p−δ with δ > 0.)Taking expectation on both sides and applying Hölder’s inequality, since q < p ∗ , it follows fromAssumptions A1–A3 that Assumption M1 holds with α < t.To prove Assumption M2, let us consider Y hl := Q hl − Q hl−1 . As above, it follows fromLemma 2.1 and Theorem 3.9 together with the triangle inequality that, for any s < t,{}|Y hl (ω)| |u hl (ω, ·) − u hl−1 (ω, ·)| H 1 (D) max |u hl (ω, ·)| q−1H 1 (D) , |u h l−1(ω, ·)| q−1H 1 (D) , 1 C 3.1(ω) C 3.8 (ω)a q−1min (ω) ‖f(ω, ·)‖ q H s−1 (D) hs l , almost surely,where the hidden constant depends on M ∗ . It follows again from Assumptions A1-A3 and Hölder’sinequality, since q < p ∗ /2, thatV[Y hl ] = E[Yh 2 l] − (E[Y hl ]) 2 ≤ E[Yh 2 l] h β l, where β < 2t.Proposition 4.3. Let Q := ‖u‖ q L 2 (D) , for some 1 ≤ q < p ∗/2. Suppose that D is a C 2 boundeddomain and that Assumptions A1–A4 hold, for some 0 < t < 1. Then Assumptions M1–M2 inTheorem 4.1 hold for any α < 2t and β < 4t. For t = 1, we get α = 2 and β = 4.Proof. This can be shown in the same way as Proposition 4.2 using Corollary 3.10 instead ofTheorem 3.9.Substituting these values into Theorem 4.1 we can get theoretical upper bounds for the ε-costsof classical and multilevel Monte Carlo in the case of log-normal fields a, as shown in Table 1.We assume here that we can obtain individual samples in optimal cost C l h −dllog(h −1l) via amultigrid solver, i.e. γ = d+δ for any δ > 0. We clearly see the advantages of the multilevel MonteCarlo method. More importantly, note that in the exponential covariance case in dimensions d > 1,the cost of MLMC is proportional to the cost of obtaining one sample on the finest grid, i.e. solvingone deterministic PDE with the same regularity properties to accuracy ε. This implies that themethod is optimal.5 Numerical ResultsIn this section, we want to confirm numerically some of the results proved in earlier sections. Allnumerical results shown are for the case of a log-normal random coefficient a(ω, x) with exponentialcovariance function (2.6). All results are calculated for a model problem in 1D. The domain Dis taken to be (0, 1), and f ≡ 1. The sampling from the random coefficient a(ω, x) is done usinga truncated KL-expansion, and as in the analysis in Section 3.3, we use the midpoint rule toapproximate the integrals in the stiffness matrix. As the quantities of interest we will just studythe simple functionals Q := ‖u‖ L 2 (D) and Q := |u| H 1 (D).17

Figure 1: Let d = 1 and σ 2 = 1 in the exponential covariance case for different choices ofthe correlation length λ. Left: Plot of ∑ n>K θ n as a function of K, i.e. the sum of the remainingKL eigenvalues when truncating after K terms. Right: The corresponding relative error∣ ∣E [ ‖u K ∗ ,h ∗‖ L 2 (D) − ‖u K,h ∗‖ L 2 (D)]∣∣ /E [ ‖u K ∗ ,h ∗‖ L 2 (D)]for problem (2.8) with K ∗ = 5000 andh ∗ = 1/2048. The dotted line has a gradient of −1.5.1 Convergence with Respect to KWe start with the results in Section 2.4. We want to show that in the exponential covariance casea relatively large number of KL-modes are necessary (even in 1D) to obtain acceptable accuraciesespecially when the correlation length is smaller than the diameter of the domain. In Figure 1 westudy the decay of the eigenvalues corresponding to the KL-expansion, as well as the convergencewith respect to K of E [ [‖u K ‖ L (D)] 2 to E ‖u‖L (D)] 2 . Since we do not know the exact solution andcannot solve the differential equation explicitly, we approximate u by u K ∗ ,h ∗ with K∗ = 5000 andh ∗ = 1/2048 and u K by u K,h ∗ with h ∗ = 1/2048.In the left figure we plot ∑ n>K θ n, i.e. the sum of the remaining eigenvalues when truncatingafter K terms. We see that after a short pre-asymptotic phase (depending on the size of λ) this sumdecays linearly in K −1 . On the right we plot ∣ ∣E [ ‖u K ∗ ,h ∗‖ L 2 (D) − ‖u K,h ∗‖ L (D)]∣∣ 2 /E [ ‖u K ∗ ,h ∗‖ L (D)] 2 ,the relative error. We see that the error decays roughly like ∑ n>K θ n and thus also linearly in K −1 .This is better than the bound for E [ ‖u K ∗ ,h ∗ − u K,h ∗‖ ∣L (D)] 2 ≥ E [ ‖u K ∗ ,h ∗‖ L 2 (D) − ‖u K,h ∗‖ L (D)]∣∣2in Proposition 2.8, which predicts at most O(K −1/2 ) convergence, and it is related to weak convergence.As shown in [5, Section 5], the weak convergence order of the PDE solution with truncatedKL-expansion can be O( ∑ n>K θ n) for certain functionals. We believe that the faster convergenceof E [ [‖u K ‖ L (D)] 2 to E ‖u‖L (D)] 2 can be shown using similar techniques.5.2 Convergence with Respect to hWe now move on to the results in the main Section 3. For the reference solution we choose againh ∗ = 1/2048 and K ∗ = 5000. Figure 2 shows the convergence of E [ |u K ∗ ,h ∗ − u K ∗ ,h| H 1 (D)]andE [ ‖u K ∗ ,h ∗ − u K ∗ ,h‖ L 2 (D)]for (2.8). In the plots, a dash-dotted line indicates O(h 1/2 ) convergenceand a dotted line indicates O(h) convergence.In both cases, we can see a short pre-asymptotic phase, which is again related to the correlationlength λ. We can see in the right plot that the H 1 –error converges with O(h 1/2 ), confirming thesharpness of the bound proven in Theorem 3.13. The L 2 –error converges linearly in h, and so thequadrature error does not seem to be dominant here.18

Figure 2: Left: Plot of E [ |u K ∗ ,h ∗ − u K ∗ ,h| H 1 (D)]versus 1/h for model problem (2.8) with d = 1,λ = 0.1 and σ 2 = 3. Right: Corresponding L 2 -error E [ ‖u K ∗ ,h ∗ − u K ∗ ,h‖ L 2 (D)]. The gradient ofthe dash-dotted (resp. dotted) line is −1/2 (resp. -1).Figure 3: Left: Plot of |E [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,h ∗‖ L 2 (D)]|, for model problem (2.8) with d = 1,λ = 0.1, σ 2 = 3, K ∗ = 5000 and h ∗ = 1/2048. Right: Corresponding plot of the varianceV [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,2h‖ L 2 (D)]. The gradient of the dotted (resp. dashed) line is −1 (resp. −2).5.3 Multilevel Monte Carlo ConvergenceWe first want to confirm Assumptions M1 and M2 in Theorem 4.1, for Q = ‖u‖ L 2 (D). In otherwords, we want to confirm the rate of decay for the quantities |E [ ‖u K ∗ ,h ∗‖ L 2 (D) − ‖u K ∗ ,h‖ L 2 (D)]|and V [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,2h‖ L 2 (D)]. Again, we choose K ∗ = 5000 and h ∗ = 1/2048 and look atthe model problem (2.8) for d = 1. The results are shown in Figure 3. A dotted line indicateslinear convergence, and a dashed line indicates quadratic convergence.For the variance V [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,2h‖ L 2 (D)](right plot), we observe the quadratic convergencepredicted by Proposition 4.3. The expected value in the left plot seems to converge slightlyfaster than the predicted linear convergence. In our experience, this faster convergence dependson the choice of model problem and quantity of interest and will need to be investigated further,since it directly affects the cost of the multilevel Monte Carlo method through the size of the ratioβ/α in the bound in Theorem 4.1.The actual performance of the standard MC and MLMC estimators in estimating Q = ‖u‖ L 2 (D)is shown in Figure 4. In the right plot the accuracy is scaled by the expected value of the quantity19

Figure 4: Left: Number of samples N l per level. Right: Plot of the cost scaled by ε −2 of theMLMC and standard MC estimators for d = 1, with λ = 0.1 and σ 2 = 3. The coarsest mesh sizein all tests is h 0 = 1/16.of interest. We see a clear advantage of the multilevel Monte Carlo method. For more numericalresults, in particular results in 2D, we refer the reader to [7].6 Conclusions and Further WorkOne of the major bottlenecks in probabilistic uncertainty quantification is the efficient numericalsolution of PDEs with random coefficients. A typical model problem that arises in subsurface flowis an elliptic PDE with log-normal coefficients. The resulting PDE is not uniformly elliptic orbounded with respect to the random variable ω and usually lacks also full regularity. In this paper,we carry out for the first time a careful finite element error analysis under these weaker assumptionsand provide optimal error estimates — with respect to the given regularity of the coefficients —for all finite moments of the finite element error (including quadrature errors). This then allowsus to rigorously analyse the convergence and the cost of multilevel Monte Carlo methods, recentlyproposed for elliptic PDEs with random coefficients, in particular in the practically important caseof log-normal coefficients with exponential covariance which has hitherto been unproved.Via duality arguments, the results in this paper are readily applicable also to other functionalsof the solution of more practical relevance. Other important aspects which we want to investigatein the future are the influence of the correlation length λ on the MLMC convergence, as well as thesuperconvergence in the expected value of certain model problems and quantities of interest (ashighlighted in the discussion of Figure 2). Another issue which will require further research is thechoice of quadrature rule. So far we have only used simple midpoint or trapezoidal rules. In futurework we would like to also explore multiscale methods and model reduction on coarser grids.ADetailed Regularity ProofIn this appendix, we give more detailed proofs of Proposition 3.1 and Lemmas 3.2 and 3.3. Theproofs follow those of Hackbusch [18, Theorems 9.1.8, 9.1.11 and 9.1.16], making explicit thedependence of all the constants that appear on the PDE coefficient a. The proof follows theclassical Nirenberg translation method and consists in three main steps.20

A.1 Step 1 – The Case D = R dProof of Lemma 3.2. In this proof we will use the norm on H s (R d ) provided by the Fourier transform,‖u‖ 2 H s (R d ) := ‖û(ξ)(1 + |ξ|2 ) s/2 ‖ L 2 (R d ), which is equivalent to the norm defined previouslyand defines the same space.For any h > 0, we define the fractional difference operator (in direction i = 1, . . . , d) bywhere e i is the ith unit vector in R d ,Rh i (v)(x) := ∑+∞ ( ) s h−s e −µh (−1) µ v(x + µhe i ),µ( s= 1 and0)µ=0( sµ)(−1) µ =−s(1 − s)(2 − s)...(µ − 1 − s).µ!Let us recall here some properties of Rh i from [18, Proof of Theorem 9.1.8]:• (Rh i )∗ (v)(x) = h −s ∑ ( )+∞µ=0 e−µh (−1) µ sµv(x − µhe i ).• For any τ ∈ R and v ∈ H τ+s (R d ),‖R i h v‖ H τ (R d ) ≤ ‖v‖ H τ+s (R d ) and ‖(R i h )∗ v‖ H τ (R d ) ≤ ‖v‖ H τ+s (R d ).(A.1)• ̂R i h v(ξ) = [(1 − e−h+iξ jh )/h] sˆv(ξ) and ̂ (Rih) ∗ v(ξ) = [(1 − e −h−iξ jh )/h] sˆv(ξ).We define for u, v ∈ H 1 (R d ) the bilinear form∫d(u, v) := A∇u∇Rh ∫R i (v) dx − A∇(Rh i )∗ u∇v dxd R∞∑( ) ∫ds= h −s e −µh (−1) µ (A(x − µhe i ) − A(x))∇u(x − µhe i )∇v dx.µ R dHence,µ=1where the hidden constant is proportional to|d(u, v)| |A| C t (R d ,S d (R)) |u| H 1 (R d ) |v| H 1 (R d ),∞∑( ) sh −s e −µh (−1) µ (µh) t ,µµ=1(which is finite, since sµ)= O(µ −s−1 ) and thus ∑ ∞µ=1 e−µh µ t−s−1 = O(h s−t ). The three spacesH 1−s (R d ) ⊂ L 2 (R d ) ⊂ H s−1 (R d ) form a Gelfand triple, so that we can deduce, using (A.1), that∫A min |(Rh i )∗ w| 2 H 1 (R d )≤ A∇(Rh i )∗ w∇(Rh i )∗ w dxR d= −d(w, (R i h )∗ w) + 〈F, R i h (Ri h )∗ w〉 H s−1 (R d ),H 1−s (R d )≤ |d(w, (R i h )∗ w)| + ‖F ‖ H s−1 (R d )‖R i h (Ri h )∗ w‖ H 1−s (R d ) |A| C t (R d ,S d (R)) |w| H 1 (R d ) |(R i h )∗ w| H 1 (R d ) + ‖F ‖ H s−1 (R d ) ‖(R i h )∗ w‖ H 1 (R d ),21

therefore we getA min ‖(R i h )∗ w‖ 2 H 1 (R d ) |A| C t (R d ,S d (R)) |w| H 1 (R d ) |(R i h )∗ w| H 1 (R d ) +and finally, using (A.1) once more,‖(R i h )∗ w‖ H 1 (R d )‖F ‖ H s−1 (R d ) ‖(R i h )∗ w‖ H 1 (R d ) + A min ‖(R i h )∗ w‖ 2 L 2 (R d ) |A| C t (R d ,S d (R)) |w| H 1 (R d ) |(R i h )∗ w| H 1 (R d ) +‖F ‖ H s−1 (R d ) ‖(R i h )∗ w‖ H 1 (R d ) + A min ‖(R i h )∗ w‖ H −1 (R d )‖(R i h )∗ w‖ H 1 (R d ),1()|A|A C t (R d ,S d (R)) |w| H 1 (R d ) + ‖F ‖ H s−1 (R d ) + ‖w‖ L 2 (R d ).minFor any 1 ≥ h > 0, since |1 − e −h−iξ ih | 2 ≥ |Im(1 − e −h−iξ ih )| 2 = e −2h sin(ξ i h) 2 ≥ e −2 sin(ξ i h) 2 , andsince sin 2 (ξh) ≥ ( 2π ξh) 2 , for all |ξ| ≤ 1/h, we have conversely thatd∑‖(Rh i )∗ w‖ 2 H 1 (R d )i=1≥=≥∫∫|ξ|≤1/h|ξ|≤1/he −2 ∫∫|ξ|≤1/h(1 + |ξ| 2 )d∑i=1| (R ̂h i )∗ w(ξ)| 2 dξd∑(1 + |ξ| 2 )1 − e −h−iξ ih∣ h ∣i=1d∑(1 + |ξ| 2 )sin(ξ i h)∣ h ∣|ξ|≤1/hi=1(1 + |ξ| 2 )|ξ| 2s |ŵ(ξ)| 2 dξ.2s2s|ŵ(ξ)| 2 dξ|ŵ(ξ)| 2 dξHence, for any 0 < h ≤ 1, we obtain‖w‖ 2 H 1+s (R d )≤∫(1 + |ξ| 2 )|ξ| 2s |ŵ(ξ)| 2 dξ +R d ∫(1 + |ξ| 2 )|ŵ(ξ)| 2 dξR dd∑≤ ‖(Rh i )∗ w‖ 2 H 1 (R d ) + ‖w‖2 H 1 (R d )and soi=1‖w‖ H 1+s (R d ) 1 ()|A|A C t (R d ,S d (R)) |w| H 1 (R d ) + ‖F ‖ H s−1 (R d ) + ‖w‖ H 1 (R d ) < +∞.minA.2 Step 2 – The Case D = R d +In order to proof Lemma 3.3 we will need the following two lemmas.Lemma A.1 ([18, Lemma 9.1.12]). Let s > 0, s ≠ 1/2, then(∥ d∑|||v||| s := ‖v‖ 2 L 2 (R d + ) + ∂v ∥∥∥2∥∂x idefines a norm on H s (R d +) that is equivalent to ‖v‖ H s (R d + ) .i=1H s−1 (R d + ) ) 1/222

Lemma A.2. Let D ⊂ R d and 0 < s < t < 1. If b ∈ C t (D) and v ∈ H s (D), then bv ∈ H s (D) and‖bv‖ H s (D) |b| C t (D) ‖v‖ L 2 (D) + |b| C 0 (D) ‖v‖ H s (D).The hidden constant depends only on t, s and d.Proof. This is a classical result, but we require again the exact dependence of the bound on b.First note that trivially ‖bv‖ L 2 (D) ≤ b max ‖v‖ L 2 (D) where b max := |b| C 0 (D). Now, for any x, y ∈ D,we have|b(x)v(x) − b(y)v(y)| 2 ≤ 2(b(x) 2 |v(x) − v(y)| 2 + v(y) 2 |b(x) − b(y)| 2 ).Continuing v by 0 on R d and denoting this extension by ṽ, this implies|b(x)v(x) − b(y)v(y)|∫∫D 2∫∫2 ‖x − y‖ d+2s dx dy ≤ 2b 2 max‖v‖ 2 H s (D) + 2 v(y) 2 |b(x) − b(y)| 2D 2 ‖x − y‖ d+2s∫∫≤ 2b 2 max‖v‖ 2 H s (D) + 8b 2 v(y) 2max‖x − y‖ d+2s + v(y) 22|b|2 dx dyC t (D)‖x − y‖ d+2(s−t)x,y∈D‖x−y‖≥1∥ ∥ )≤ 2b 2 max‖v‖ 2 H s (D)(8b + 2 1 ‖z‖≥1 ∥∥∥Lmax ∥‖z‖ d+2s + 2|b| 2 1 ‖z‖≤1 ∥∥∥LC t (D) ∥1 (R d ) ‖z‖ d+2(s−t) ‖ṽ‖ 2 L 2 (R d )1 (R d ) b 2 max‖v‖ 2 H s (D) + |b|2 C t (D) ‖v‖2 L 2 (D) .Proof of Lemma 3.3. First we continue the solution w by 0 on R d \ R d + and denote the extension˜w ∈ H 1 (R d ). Take 1 ≤ i ≤ d − 1. Similarly to the previous section, we define for u, v ∈ H 1 (R d )∫∫d(u, v) := A∇u∇Rh i (v) dx − A∇(Rh i )∗ u∇v dxand deduce again thatR d +R d +|d(u, v)| |A| C t (R d + ,S d(R)) |u| H 1 (R d + ) |v| H 1 (R d + ) .We now note that, since i ≠ d, (R i h )∗ w ∈ H 1 0 (Rd +) and (R i h )∗ ˜w ∈ H 1 (R d ) is equal to the continuationby 0 on R d \ R d + of (R i h )∗ w. We deduce, similarly to the proof in Section A.1 using (A.1), that‖(R i h )∗ ˜w‖ H 1 (R d )1()|A|A C t (R d min + ,S d(R)) |w| H 1 (R d + ) + ‖F ‖ H s−1 (R d + ) + ‖w‖ H 1 (R d + ) =: B(w).(Note that we added |w| H 1 (R d + ) to the bound to simplify the notation later.) Hence, by the sametoken as in the previous section, we get∫(1 + |ξ| 2 )(|ξ 1 | 2 + ... + |ξ d−1 | 2 ) s |̂˜w(ξ)| 2 dξ B(w) 2 .(A.2)R dIn particular, this implies that, for 1 ≤ i ≤ d and 1 ≤ j ≤ d − 1, we have∫ ∣ ∣∣∣∣ ̂∂2 ˜w(ξ)R d ∂x i ∂x j ∣2(1 + |ξ| 2 ) s−1 dξ =∫R d |ξ i | 2 |ξ j | 2 |̂˜w(ξ)| 2 (1 + |ξ| 2 ) s−1 dξ B(w) 2 ,23

∥∂which means that2 ˜w∂x i ∂x j∈ H s−1 (R d ) and ∥ ∂2 ˜w ∥∥H∂x i ∂x j B(w). In particular, for all (i, j) ≠s−1 (R d ) ∥∂(d, d), this further implies that2 w∂x i ∂x j∈ H s−1 (R d +) and that ∥ ∂2 w ∥∥H∂x i ∂x j B(w). Usings−1 (R d + )Lemma A.1 we deduce that ∂w∂x j∈ H s (R d +) and that∥ ∂w ∥∥∥H∥ B(w), for all 1 ≤ j ≤ d − 1. (A.3)∂x j s (R d + )∥ It remains to bound ∥ ∂w ∥∥H∂x d, which is rather technical. To achieve it we will use the PDE (3.1),s (R d + )Lemma A.2 and the following result.Lemma A.3. For almost all x d ∈ R, we have ∂ ˜w∂x d(., x d ) ∈ H s (R d−1 ) and∫∥ ∂ ˜w∫(., x d ) ∥ 2 ∣ R ∂x dx d H s (R d−1 d = (1 + |ξ ′ | 2 ) s |ξ d | 2 ∣∣̂˜w(ξ) ∣ 2 dξ B(w) 2 .) R dProof. This follows from Fubini’s theorem and Plancherel’s formula, together with (A.2).From this we deduce that ∂w∂x d(., x d ) ∈ H s (R d−1 ), for almost all x d ∈ R + , and that∫ ∥ ∥∥ ∂w(., x d ) ∥ 2R +∂x dx d H s (R d−1 d B(w) 2 .)∂wLet 1 ≤ i ≤ d − 1. Using Lemma A.2 we deduce that A id ∂x d(., x d ) ∈ H s (R d−1 ), for almost allx d ∈ R + , and that( )∂w∥ A id (., x d )∂x d∥ |A id (., x d )| C ∂wt (R d−1 ) ∥ (., x d )H s (R d−1 )∂x d∥L 2 (R d−1 )+ ‖A id (., x d )‖ C ∂w0 (R d−1 ) ∥ (., x d ) ∥ .∂x d∥H s (R d−1 )Therefore, since by definition ‖A id ‖ C 0 (R d ) ≤ A max , we get∫ ∥ ∥∥Aid ∂w∥ ∥∥2R +∂x dx d H s (R d−1 d |A id | 2) C t (R d + ) |w|2 H 1 (R d + ) + A2 max B(w) 2 A 2 max B(w) 2 .∂∂x iSince is linear continuous from H 1−s (R d−1 ) to H −s (R d−1 ) (cf. [18, Remark 6.3.14(b)]) we can( )∂ ∂wdeduce from this that∂x iA id ∂x d∈ H s−1 (R d +) and that( )∥ ∂ ∂w ∥∥∥H∥ A id A max B(w), for all 1 ≤ i ≤ d − 1. (A.4)∂x i ∂x d s−1 (R d + )To see this take ϕ ∈ D(R d +). Then〈 ( ) 〉∣ ∂ ∂w∣∣∣∣ A∣id , ϕ=∂x i ∂x d D ′ (R d + ),D(Rd + )≤≤∥ A ∂wid (x ′ , x d ) ∂ϕ(x ′ , x d ) ∥∂x d ∂x i ∥ ∥ A ∂w ∥∥∥L ∂ϕid∂x d∥ (x ′ , x d ) ∥2 (R + ,H s (R d−1 ))∂x i ∥ ∥ A ∂w ∥∥∥Lid∂x d 2 (R + ,H s (R d−1 ))∥L 2 (R + ,H s (R d−1 ))∥L 2 (R + ,H −s (R d−1 ))‖ϕ‖ L 2 (R + ,H 1−s (R d−1 )).24

( )∂ ∂wUsing (A.3) and Lemma A.2, we deduce in a similar way that∂x iA ij ∂x j∈ H s−1 (R d +) and that( )∥ ∂ ∂w ∥∥∥H∥ A ij A max B(w), for all 1 ≤ i ≤ d and 1 ≤ j ≤ d − 1. (A.5)∂x i ∂x j s−1 (R d + )We can now use the PDE (3.1) to get(a similar)bound for (i, j) = (d, d). Since F ∈ H s−1 (R d +),∂ ∂wit follows from (A.4) and (A.5) that∂x dA dd ∂x d∈ H s−1 (R d +) and that( )∥ ∂ ∂w ∥∥∥H∥ A dd A max B(w) + ‖F ‖∂x d ∂x H s−1 d s−1 (R d + ) (R d + ) A max B(w) .Analogously to (A.4) we can prove that( )∥ ∂ ∂w ∥∥∥H∥ A dd∂x i ∂x d s−1 (R d + ) A max B(w), for all 1 ≤ i ≤ d − 1.∂wHence, we can finally apply Lemma A.1 to get that A dd ∂x d∥ ∥ A ∂w ∥∥∥2d∑( )∥ dd∂ ∂w ∥∥∥2∂x d∥ A dd∂x i ∂x dH s (R d + )i=1H s−1 (R d + ) +∈ H s (R d +) and that∥ ∥ A ∂w ∥∥∥2dd A 2 max B(w) 2 .∂x d L 2 (R d + )∂wBy applying Lemma A.2 again, this time with b := 1/A dd and v := A dd ∂x d, we deduce that∂w∂x d∈ H s (R d +) and that∥ ∣ ∥ ∥ ∂w ∥∥∥H ∥1 ∣∣∣C ∥∥∥ ∂w ∥∥∥L∂x d∣A dd + 1 ∥ ∥ ∥∥∥ ∂w ∥∥∥HA dd s (R d + ) A dd t (R d + ) ∂x d 2 (R d + ) A min ∂x d s (R d + )|A dd | C t (R d + )A 2 minA max |w| H 1 (R d + ) + 1A min∥ ∥∥∥A dd∂w∂x d∥ ∥∥∥H s (R d + ) A maxA minB(w) .To finish the proof we use this bound together with (A.3) and apply once more Lemma A.1 toshow that w ∈ H 1+s (R d +) and‖w‖ H 1+s (R d + ) A )max(|A|A 2 C t (R d +min,S d(R)) |w| H 1 (R d + ) + ‖F ‖ H s−1 (R d + ) + A max‖w‖A H 1 (R d ).min+A.3 Step 3 – The Case D BoundedWe can now prove Proposition 3.1 using Lemmas 3.2 and 3.3 in two successive steps. We recallthat D was assumed to be C 2 . Let (D i ) 0≤i≤m be a covering of D such that the (D i ) 0≤i≤m are openand bounded, D ⊂ ∪ m i=0 D i, ∪ m i=1 (D i ∩ ∂D) = ∂D, D 0 ⊂ D.Let (χ i ) 0≤i≤m be a partition of unity subordinate to this cover, i.e. we have χ i ∈ C ∞ (R d , R + )with compact support supp(χ i ) ⊂ D i , such that ∑ pi=0 χ i = 1 on D. We denote by u the solutionof (2.3) and split it into u = ∑ pi=0 u i, with u i = uχ i . We treat now separately u 0 and then u i ,1 ≤ i ≤ p, using Section A.1 and A.2, respectively.Lemma A.4. u 0 belongs to H 1+s (D) and‖u 0 ‖ H 1+s (D) ‖a‖ C t (D)a 2 ‖f‖ H s−1 (D) .min25

Proof. Since supp(u 0 ) ⊂ D 0 , we have that u 0 ∈ H0 1 (D) and it is the weak solution of the newequation −div(a∇u 0 ) = F on D, whereF := fχ 0 + a∇u · ∇χ 0 + div(au∇χ 0 ) on D.To apply Lemma 3.2 we will now extend all terms to R d , but continue to denote them the same.The terms u 0 and fχ 0 + a∇u · ∇χ 0 can both be extended by 0. Their extensions will belongto H 1 (R d ) and H s−1 (R d ), respectively. It follows from Lemma A.2 that au ∈ H s (D) and so ifwe continue au∇χ 0 by 0 on R d , the extension belongs to H s (R d ), since supp(χ 0 ) is compact inD. Using the fact that div is linear and continuous from H s (R d ) to H s−1 (R d ) (cf. [18, Remark6.3.14(b)] and the proof of (A.4) above), we can deduce that the divergence of the extension ofau∇χ 0 is in H s−1 (R d ), leading to an extension of F on R d , which belongs to H s−1 (R d ).Let ψ ∈ C ∞ (R d , [0, 1]) such that ψ = 0 on D 0 and ψ = 1 on ˜D c , where ˜D is an open set suchthat D 0 ⊂ ˜D and ˜D ⊂ D. We use the following extension of a from D 0 to all of R d :{ a(x)(1 − ψ(x)) + amin ψ(x), if x ∈ D,a(x) :=a min ψ(x), otherwise.This implies that a ∈ C t (R d ), and for any x ∈ R d , a min ≤ a(x) ≤ a(x) and |a| C t (R d ) ‖a‖ C t (D) .Using these extensions, we have that −div(a∇u 0 ) = F in D ′ (R d ). Indeed, for any v ∈ D(R d ),∫∫a(x)∇u 0 (x)∇v(x) = a(x)∇u 0 (x)∇v(x) for any v ∈ D(R d ),R d Dsince supp(u 0 ) is included in the open bounded set D 0 , which implies that ∇u 0 = 0 on D c 0 and a = aon D 0 . Since u ∈ H 1 0 (D), we have by Poincaré’s inequality that ‖u‖ L 2 (D) |u| H 1 (D). Therefore itfollows from Lemma 2.1 that|u 0 | H 1 (R d ) ≤ |u| H 1 (D)‖χ 0 ‖ ∞ + ‖u‖ L 2 (D)‖∇χ 0 ‖ ∞ ‖f‖ H s−1 (D)a min.Since χ 0 ∈ C ∞ (R d ), using Lemma A.2 and the linearity and continuity of div from H s (R d ) toH 1−s (R d ) we further get‖F ‖ H s−1 (R d ) ≤ ‖fχ 0 ‖ H s−1 (R d ) + ‖a∇u · ∇χ 0 ‖ H s−1 (R d ) + ‖div(au∇χ 0 )‖ H s−1 (R d )‖f‖ H s−1 (D) + a max |u| H 1 (D) + |a| C t (D)‖u‖ L 2 (D) + a max ‖u‖ H s (D)‖a‖ C t (D)‖f‖a H s−1 (D).minWe can now apply Lemma 3.2 with A = aI d and w = u 0 to show that u 0 ∈ H 1+s (R d ) and‖u 0 ‖ H 1+s (R d ) 1 ()|a|a C t (D) |u 0| H 1 (R d ) + ‖F ‖ H s−1 (R d ) + ‖u 0 ‖ H 1 (R d ) ‖a‖ C t (D)min a 2 ‖f‖ H s−1 (D) .minThe hidden constant depends on the choices of χ 0 and ψ and on the constant in Poincaré’s inequality,which depends on the shape and size of D, but not on a.Let us now treat the case of u i , 1 ≤ i ≤ m.Lemma A.5. For 1 ≤ i ≤ m, u i ∈ H 1+s (D) and‖u i ‖ H 1+s (D) a max‖a‖ C t (D)a 3 ‖f‖ H s−1 (D).min26

Proof. Similarly to the proof of the previous Lemma, u i ∈ H 1 0 (D ∩ D i) is the weak solution of anew problem −div(a∇u i ) = g i in D ′ (D ∩ D i ) withg i := fχ i + a∇u · ∇χ i + div(au∇χ i ).As in Lemma A.4, since again div is linear and continuous from H s to H s−1 , we can establish thatg i ∈ H s−1 (D ∩ D i ) and‖g i ‖ H s−1 (D∩D i ) ‖a‖ C t (D)a min‖f‖ H s−1 (D).Now let Q = {(y ′ , y d ) ∈ R d−1 ×R : |y ′ | < 1 and |y d | < 1}, Q 0 = {(y ′ , y d ) ∈ R d−1 ×{0}| ‖y ′ ‖ < 1}and Q + = Q ∩ R d +. For 1 ≤ i ≤ p, let α i be a bijection from D i to Q such that α i ∈ C 2 (D i ),αi −1 ∈ C 2 (Q), α i (D i ∩ D) = Q + and α i (D i ∩ ∂D) = Q 0 .For all y ∈ Q + , we define w i (y) := u i (αi−1 (y)) ∈ H0 1(Q +) with ∇w i (y) = Ji−t (y)∇u i (αi−1 (y)),where J i (y) := Dα i (αi −1 (y))) is the Jacobian of α i . Furthermore, for x ∈ D i ∩ D and ϕ ∈ H0 1(Q +),we define v(x) := ϕ(α i (x)). Then v ∈ H0 1(D i ∩ D) and ∇v(αi −1 (y)) = Ji t(y)∇ϕ(y), for all y ∈ Q +,so that∫∫whereD i ∩Da(x)∇u i (x) · ∇v(x) dx =We define F i ∈ H s−1 (Q + ) by=∫a(αi−1Q +(y)) ∇u i (α −1iQ +A i (y) ∇w i (y) · ∇ϕ(y) dy ,A i (y) := a(αi−1 (y)) |detJ i (y))| −1 (J i Ji t )(y) ∈ S d (R).(y)) · ∇v(α −1 (y)) |detJ i (y))| −1 dy〈F i , ϕ〉 H s−1 (Q + ),H 1−s0 (Q + ):= 〈g i , ϕ ◦ α i 〉 H s−1 (D i ∩D),H 1−s0 (D i ∩D), for all ϕ ∈ H1−s 0 .Indeed, since we assumed that α i and αi−1 are in C 2 , we have ϕ ◦ α i ∈ H0 1−s (D i ∩ D) and moreover‖ϕ ◦ α i ‖ H 1−s (D i ∩D) ‖ϕ‖ H 1−s (Q + ) (cf. [18, Theorems 6.2.17 and 6.2.25(g)]), which implies thatF i ∈ H s−1 (Q + ) and‖F i ‖ H s−1 (Q + ) ‖g i ‖ H s−1 (D∩D i ).We finally get that v i ∈ H 1 0 (Q +) solves∫Q +A i ∇v i · ∇ϕ dy = 〈F i , ϕ〉 H s−1 (Q + ),H 1−s0 (Q + )for all ϕ ∈ H 1 0 (Q + ).In order to apply Lemma 3.3 we check first that A i ∈ C t (Q + , S d (R)) and that it is coercive, andthen define an extension of A i to R d +. Recalling that α i is a C 2 –diffeomorphism from D i ∩ D toQ + , with α −1i∈ C 2 (Q + ), we have for any y ∈ Q + and ξ ∈ R d :• Coercivity:• Boundedness:A i (y)ξ · ξ = a(α −1i(y))|detJ i (y)| −1 |J t i (y)ξ|2 a min |ξ| 2 . Hence A min a min .A max := ‖A i ‖ C 0 (Q + ,S d (R)) = max a(x) ‖|detJ i | −1 J i Ji t ‖ Cx∈D i ∩D0 (Q + ,S d (R)) a max .i• Regularity:A i ∈ C t (Q + , S d (R)) and‖A i ‖ C t (Q + ,S d (R))≤ a max ‖|detJ i | −1 J i J t i ‖ C t (Q + ,S d (R))+ |a| C t (D) ‖|detJ i| −1 J i J t i ‖ C 0 (Q + ,S d (R)) ‖a‖ C t (D) .27

We now extend A i to R d +. Since we assumed that supp(χ i ) is compact in D i , we can choose Q iand ˜Q i such that supp(v i ) ⊂ Q i ⊂ Q i ⊂ ˜Q i ⊂ ˜Q i ⊂ Q and consider ψ ∈ C ∞ (R d , [0, 1]) such thatψ = 0 on Q i and ψ = 1 on ˜Qci . We define the extension Ai of A i on R d + by{Ai (x)(1 − ψ(x)) + aA i (x) :=min ψ(x)I d if x ∈ Q +a min ψ(x)I d if x ∈ Q c +Analogously to the case of a in Lemma A.4, we can deduce that, for any y ∈ R d + and ξ ∈ R d ,A i (y)ξ ·ξ a min |ξ| 2 , A max = ‖A i ‖ C 0 (R d + ,S d(R)) a maxand ‖A i (y)‖ C t (R d + ,S d(R)) ‖a‖ C t (D) . (A.6)We now define an extension of F i on R d +. Note again that we can choose an open set G i such thatsupp(F i ) ⊂ G i ⊂ G i ⊂ Q and extend F i to all of R d + such that ‖F i ‖ H s−1 (R d + ) ‖F i‖ H s−1 (Q + ). Finallywe continue v i by 0 on R d +, which yields v i ∈ H 1 0 (Rd +). Moreover, since A i = A i on supp(v i ) ⊂ Q i ,v i is then the weak solution on R d + of−div(A i (x)∇v i (x)) = F i (x),which enables us to apply Lemma 3.3 and to obtain that v i ∈ H 1+s (R d +) and‖v i ‖ H 1+s (R d + ) A )max(|AA 2 i | C t (R d +min,S d(R)) |v i| H 1 (R d + ) + ‖F i‖ H s−1 (R d + )+ A maxA min‖v i ‖ H 1 (R d + ) .Recalling that u i (x) = v i (α i (x)) for any x ∈ D ∩ D i and using the bounds in (A.6), as well as thetransformation theorem [18, Theorem 6.2.17], we finally get‖u i ‖ H 1+s (D) a ()maxa 2 ‖a‖ C t (D) ‖u‖ H 1 (D∩D i ) + ‖g i ‖ H s−1 (D∩D i ) + a max‖u‖mina H 1 (D)mina max ‖a‖ C t (D)‖f‖ H s−1 (D) .a 3 min∑The result in Proposition 3.1 follows directly from Lemmas A.4 and A.5, if we recall that u = m u i .References[1] I. Babuška, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partialdifferential equations with random input data. SIAM J. Numer. Anal., 45(3):1005–1034, 2007.[2] I. Babuška, R. Tempone, and G. E. Zouraris. Galerkin finite element approximations ofstochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42(2):800–825, 2004.[3] A. Barth, C. Schwab, and N. Zollinger. Multi–level Monte Carlo finite element method forelliptic PDE’s with stochastic coefficients. SAM Research Reports 2010-18, ETH Zürich, 2010.[4] S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods, volume 15of Texts in Applied Mathematics. Springer, New York, third edition, 2008.[5] J. Charrier. Strong and weak error estimates for the solutions of elliptic partial differentialequations with random coefficients. Technical Report HAL:inria-00490045, Hyper Articles enLigne, INRIA, 2010. Available at http://hal.inria.fr/inria-00490045/en/.i=028

[6] P. G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in AppliedMathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.[7] K.A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup. Multilevel Monte Carlo methodsand applications to elliptic PDEs with random coefficients. Computing and Visualization inScience, 2011. accepted for publication.[8] G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions, volume 44 of Encyclopediaof Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.[9] M. Dashti and A. Stuart. Uncertainty quantification and weak approximation of an ellipticinverse problem. Technical Report arXiv:1102.0143, arXiv.org, 2010. Submitted to SIAMJournal on Numerical Analysis.[10] G. de Marsily, F. Delay, J. Goncalves, P. Renard, V. Teles, and S. Violette. Dealing withspatial heterogeneity. Hydrogeol. J, 13:161–183, 2005.[11] P. Delhomme. Spatial variability and uncertainty in groundwater flow param- eters, a geostatisticalapproach. Water Resourc. Res., pages 269–280, 1979.[12] P. Frauenfelder, C. Schwab, and R. A. Todor. Finite elements for elliptic problems withstochastic coefficients. Comput. Methods Appl. Mech. Engrg., 194(2-5):205–228, 2005.[13] J. Galvis and M. Sarkis. Approximating infinity-dimensional stochastic Darcy’s equationswithout uniform ellipticity. SIAM Journal on Numerical Analysis, 47:3624–3651, 2009.[14] R. G. Ghanem and P. D. Spanos. Stochastic finite elements: a spectral approach. Springer-Verlag, New York, 1991.[15] M. B. Giles. Improved multilevel Monte Carlo convergence using the Milstein scheme. volume256 of Monte Carlo and Quasi-Monte Carlo methods 2006, pages 343–358. Springer, 2007.[16] M. B. Giles. Multilevel Monte Carlo path simulation. Operations Research, 256:981–986, 2008.[17] C. J. Gittelson. Stochastic Galerkin discretization of the log-normal isotropic diffusion problem.Math. Models Methods Appl. Sci., 20(2):237–263, 2010.[18] W. Hackbusch. Elliptic differential equations: Theorey and numerical treatment, volume 18 ofSpringer Series in Computational Mathematics. Springer-Verlag, Berlin, 2010.[19] S. Heinrich. Multilevel Monte Carlo methods. volume 2179 of Lecture notes in ComputerScience, pages 3624–3651. Springer, Philadelphia, PA, 2001.[20] S. Zhang. Analysis of finite element domain embedding methods for curved domains usinguniform grids. SIAM Journal on Numerical Analysis, 46:2843–2866, 2008.29